The present invention relates to optical filters and, more particularly, to a phase-only filter that enables the generation of any desired illumination pattern.
A generalized optical system is shown in FIG. 1. Incoherent radiation of wavelength .lambda. is provided at an input plane 10, and passes through an imaging system of focal length F, represented schematically by a lens 12, to an output plane 14. Input plane 10 is a distance U from lens 12. Output plane 14 is a distance V from lens 12. The distances U, V, and F satisfy the relationship ##EQU1## FIG. 1 also shows the Cartesian coordinate system used herein. z is the direction of light propagation. x is perpendicular to z, in the plane of FIG. 1. y is perpendicular to both x and z, and points out of the plane of FIG. 1 at the reader.
The optical system of FIG. 1 is characterized by a coherent transfer function (CTF) H(.function..sub.x,.function..sub.y) which is related to the aperture transmittance function P(x,y) of lens 12 by the relationship EQU H(.function..sub.x,.function..sub.y)=P(-.lambda.V.function..sub.x,-.lambda. V.function..sub.y) (2)
h(x,y), the impulse response of the CTF, is the inverse Fourier transform of the CTF: ##EQU2## The optical transfer function (OTF) is the normalized transfer function of the system for incoherent illumination, defined as: ##EQU3## where the normalizing constant k is: ##EQU4## See, for example, J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, San Francisco (1968), pp. 102-130; M. J. Beran and G. B. Parrent, Theory of Partial Coherence, Prentice-Hall Inc., Englewood Cliffs N.J. (1964); L. Mandel and E. Wolf, "Coherence properties of optical fields", Rev. Mod. Phys. vol. 37 p. 231 (1965); and P. S. Considine, "Effects of coherence on imaging system", J. Opt. Soc. Am. vol. 56 p. 1001 (1996). The output obtained for the case of incoherent illumination is: ##EQU5## where I.sub.do and I.sub.gi are the output and input intensities respectively. Equivalently, the OTF of the optical system is the ratio of the Fourier transforms of the output and input intensities: ##EQU6## where "" represents a Fourier Transform operation.
Using equations (2), (4) and (5), the expression for the OTF may be rewritten ##EQU7## Equation (8) yields some of the physical properties of the OTF: ##EQU8##
Given a particular input intensity profile I.sub.gi and imaging system, a desired output intensity profile I.sub.do can be obtained by putting a suitable filter 16 between lens 12 and output plane 14, as shown in FIG. 2, thereby forcing the optical system to have the corresponding OTF as expressed by equation (7). Filter 16 typically functions by attenuating the light passing through lens 12. Thus, the energy of the output beam is less than the energy of the input beam. This is undesirable in many applications, for example in optical computing, in which many imaging systems may be cascaded, and the intensity of the ultimate output light may be too low to be of practical use. If a non-attenuating, all-phase filter, that would function as a diffractive optical element, could be designed and constructed, the output intensity would be reduced from the input intensity to a smaller extent than is the case with an attenuation filter. In fact, the output intensity would be substantially the same as the input intensity. Devices incorporating such filters would function with essentially no attenuation losses in the filters.
There is thus a widely recognized need for, and it would be highly advantageous to have, a phase only filter corresponding to a desired optical transfer function.